Stimulated Brillouin Scattering (SBS) limits the maximum optical power for narrow-spectrum signals in fiber optic systems. As such, SBS suppression is crucial for the realization of very narrow linewidth systems, such as systems of <100 MHz, with power-length products that can excite a significant amount of Brillouin Scattering (SBS), depending on the application.
Aside from employing large mode area (LMA) fibers, a number of methods exist that can be exploited to suppress SBS. For example, according to one method the laser spectrum is broadened so that the signal's effective linewidth is greater than that of the Brillouin gain spectrum (BGS). Alternatively, in another method, BGS can be broadened, consequently decreasing the peak gain, by varying the core size in a drawn fiber, varying the index of refraction, and inducing stresses in the fiber.
SBS is a well-known interaction between an acoustic wave and the optical field in fiber. In general, the scattering amplitude can be found from a volume integral
      D          p      ,      q        =            ∫      Vol        ⁢                            E          →                p        *            ⁢      δ      ⁢                        ɛ          →                          p          ,          q                    ⁢                        E          →                iq            ⁢                          ⁢              ⅆ        V            where p,q=r,φ,z are the cylindrical coordinates, {right arrow over (E)} is the electric field, and δ{right arrow over (∈)} are the dielectric perturbations resulting from the acoustic strain. Taking the optical field component Ez to be zero in the fiber, the non-zero contributions to the scattering amplitude are Drr, Dφφ, and Drφ (=Dφr). The relevant dielectric perturbations (δ∈p,q) are functions of the acoustic strain fields (Sp,q) and are expressed asδ∈rr=−∈0n4(p11Srr+p12Sφφ+p12Szz)δ∈φφ=−∈0n4(p11Sφφ+p12Srr+p12Szz)δ∈φr=−∈0n42p44Sφr where n is the index of refraction and ∈0 is the permittivity of free space. The photoelastic constants, p, are provided, for example, for vitreous silica at λ=632.8 nm as p11=0.121, p12=0.271, and p44=−0.075. The generalized form of the acoustic strain field can be written as a function of the displacement vector {right arrow over (u)}
      S    pq    =            1      2        ⁢          (                                    ∂                          u              q                                            ∂            p                          +                              ∂                          u              p                                            ∂            q                              )      
In general, the components of {right arrow over (u)} are coupled and can be found from a generalized damped acoustic wave equation,
            ρ      ⁢                          ⁢              u        ¨              -                  ∇        _            ⁢              ·                  [                                                                      c                  _                                _                            ⁢                                                ∇                  _                                ⁢                u                                      +                                                            η                  _                                _                            ⁢                                                ∇                  _                                ⁢                                  u                  .                                                              ]                      =            -              1        2              ⁢                  ∇        _            ⁢              ·                  [                                                    γ                _                            _                        ⁢                          E              k                        ⁢                          E              l                                ]                    where the electrostrictive coefficients are given by a fourth rank tensor in γ, and the damping term, ρ, is a tensor of rank four. Finally, c is the rank-four elastic modulus tensor. A damped wave equation is considered since at the acoustic frequencies involved in SBS (˜10-20 GHz) the acoustic wave is heavily damped. However, a few common approximations can be made to the generalized damped acoustic wave equation above, which simplify finding mode solutions for the equation. The first is to assume that the electrostrictive force term,
            -              1        2              ⁢                  ∇        _            ⁢              ·                  [                                    γ              ijkl                        ⁢                          E              k                        ⁢                          E              l                                ]                      ,can be neglected. Second, one may assume that the material damping coefficient ρ in the generalized damped acoustic wave equation is zero. An exponential decay term originating from material damping is then appended to the resulting mode solutions. Both of these quantities, i.e. the damping and force terms, are changing in the radial direction within the fiber. With the generalized damped acoustic wave equation simplified, the general solutions U can be taken to be a superposition of solutions, φ and Ψ, of two equations for the longitudinal and transverse waves, respectively, as followsU=∇φ+∇×Ψ(∇t+h2)φ=0(∇t=k2)Ψ=0where h and k are complex propagation constants. We define h and k as in any layer i to be
            h      i        =                            ±                                    (                                                2                  ⁢                  π                  ⁢                                                                          ⁢                                      v                    B                                                                    V                                      L                    i                                                              )                        2                          ∓                  β          2                                k      i        =                            ±                                    (                                                2                  ⁢                  π                  ⁢                                                                          ⁢                                      v                    B                                                                    V                                      S                    i                                                              )                        2                          ∓                  β          2                    where vB is the acoustic frequency, VL is the material longitudinal acoustic velocity and VS is the material shear velocity. In these equations β is the propagation constant of the acoustic wave. The correct sign will depend on which Bessel functions are solutions (J, Y or I, K) in any given layer with consideration to the acoustic velocity. The acoustic velocities in any given layer i are functions of the Lamé constants (λi and μi) as
            V              L        i              =                                        λ            i                    +                      2            ⁢                          μ              i                                                ρ          i                                V              S        i              =                            μ          i                          ρ          i                    
To determine the acoustic eigenmodes of an acoustical fiber, the general solution U above is solved subject to the typical boundary conditions. First, the displacement vector should be continuous at any interface. Second, the two shear stresses at the interface and the normal compressional stress must be continuous at any boundary. This introduces only two additional equations on top of the displacement vector boundary conditions.
In the SBS interaction, the dominant displacement vector component is uz. This makes sense since SBS is known to result from a longitudinally varying acoustically-induced Bragg grating. Further, the analysis in “Backward Collinear Guided-Wave-Acousto-Optic Interactions in Single-Mode Fibers,” by C.-K Jen and N. Goto, J. Lightwave Technol. 7, 2018-2023 (1989), showed that Drr is the dominant scattering amplitude, and thus, p12 (on Szz) plays the most significant role in determining the Brillouin gain. For completeness, the four requisite boundary conditions at an interface r=a between any regions 1 and 2 are provided in the following boundary equations:
                    u                  r          ⁢                                          ⁢          1                    ⁡              (                  r          =          a                )              =                  u                  r          ⁢                                          ⁢          2                    ⁡              (                  r          =          a                )                                u                  z          ⁢                                          ⁢          1                    ⁡              (                  r          =          a                )              =                  u                  z          ⁢                                          ⁢          2                    ⁡              (                  r          =          a                )                                                      T            rr                    =                    ⁢                      T            1                                                        =                    ⁢                                                    (                                                      λ                    1                                    +                                      2                    ⁢                                          μ                      1                                                                      )                            ⁢                                                ∂                                      u                                          r                      ⁢                                                                                          ⁢                      1                                                                                        ∂                  r                                                      ⁢                          |                              r                =                a                                      ⁢                                          +                                  λ                  1                                            ⁢                                                u                                      r                    ⁢                                                                                  ⁢                    1                                                  r                                      ⁢                          |                              r                =                a                                      ⁢                                          +                                  λ                  1                                            ⁢                                                ∂                                      u                                          z                      ⁢                                                                                          ⁢                      1                                                                                        ∂                  z                                                      ⁢                          |                              r                =                a                                                                                  =                    ⁢                                                    (                                                      λ                    2                                    +                                      2                    ⁢                                          μ                      2                                                                      )                            ⁢                                                ∂                                      u                                          r                      ⁢                                                                                          ⁢                      2                                                                                        ∂                  r                                                      ⁢                          |                              r                =                a                                      ⁢                                          +                                  λ                  2                                            ⁢                                                u                                      r                    ⁢                                                                                  ⁢                    2                                                  r                                      ⁢                          |                              r                =                a                                      ⁢                                          +                                  λ                  2                                            ⁢                                                ∂                                      u                                          z                      ⁢                                                                                          ⁢                      2                                                                                        ∂                  z                                                      ⁢                          |                              r                =                a                                                                                                  T            rz                    =                    ⁢                      μ            ⁢                                                  ⁢                          S              rz                                                                    =                    ⁢                      μ            ⁢                                                  ⁢                          S              5                                                                    =                    ⁢                                                    μ                1                            ⁡                              (                                                                            ∂                                              u                                                  r                          ⁢                                                                                                          ⁢                          1                                                                                                            ∂                      z                                                        +                                                            ∂                                              u                                                  z                          ⁢                                                                                                          ⁢                          1                                                                                                            ∂                      r                                                                      )                                      ⁢                          |                              r                =                a                                                                                  =                    ⁢                                                    μ                2                            ⁡                              (                                                                            ∂                                              u                                                  r                          ⁢                                                                                                          ⁢                          2                                                                                                            ∂                      z                                                        +                                                            ∂                                              u                                                  z                          ⁢                                                                                                          ⁢                          2                                                                                                            ∂                      r                                                                      )                                      ⁢                          |                              r                =                a                                                        The Lamé constants λi should not be confused with the optical wavelength λ.
In terms of solutions to the generalized damped acoustic wave equation, a scalar-wave part leads to longitudinal-wave components while a vector-wave part leads to shear waves. The general solution to the wave equation, general solution U above, (for the l=0 modes) for the three (cylindrical) components of the displacement vector areur(r)=AhX′l(hr)+BβZ′l(kr)uφ(r)=0uz(r)=−jAβXl(hr)+jBkZl(kr)where X and Z are linear combinations of the normal Bessel functions (J, Y, I, and K), paying particular attention to the physicality of the problem. In particular, when the eigenmode solution has an acoustic velocity greater than the material in a particular layer, the solutions are combinations of the Bessel functions of the first (J) and second kinds (Y) in that layer. When the eigenmode solution has an acoustic velocity that is less than the material, then combinations of the modified Bessel functions of the first (I) and second (K) kinds. A and B represent complex coefficients (system unknowns) which determine the characteristic matrix, and j=(−1)1/2. The primes are derivatives respect to the whole argument and not r alone.
The eigenvalues obtained from the characteristic matrix determined by subjecting the three (cylindrical) components of the displacement vector to the boundary equations above are the acoustic frequencies (νa) for each eigenmode of the system since the propagation constant is fixed to
      2    ⁢    π    ⁢                  ⁢    n    λvia the Bragg condition, where n is the index of the optical mode. In this case, since there are three boundaries, the characteristic matrix is 12×12.
Finally, the observed Brillouin gain coefficient is proportional to |Dp,q|2. However, if the volume integral above is normalized with respect to coefficients and optical intensity and acoustic field, then we may determine a BGS from
      ∑    m    ⁢          ⁢                    g        B            ⁡              (                  v                      a            m                          )              ⁢                  (                  Δ          ⁢                                          ⁢                                    v                              B                m                                      /            2                          )                                          (                          v              -                              v                                  a                  m                                                      )                    2                +                              (                          Δ              ⁢                                                          ⁢                                                v                                      B                    m                                                  /                2                                      )                    2                      ⁢          Γ      m      where the Brillouin spectral width ΔνB (FWHM) may be different for each acoustic mode with eigenfrequency νa. Acquiring values that are ≦1, Γ is the square of a normalized scattering the volume integral, and is unique for each acoustic mode m.
In order to control the acoustic and refractive index properties of the core, Table I (below) shows the effect of some common fiber dopants on the acoustic index and optical refractive index (no) when added to pure silica (assuming na=1 for pure silica). In Table I, RE represents Rare Earth.
Aside from using large mode area (LMA) fibers, numerous methods exist that can be exploited to suppress SBS. First, one can broaden the laser spectrum so that the signal's effective linewidth is greater than that of the Brillouin gain spectrum (BGS). Alternatively, one can broaden the BGS, consequently decreasing the peak gain, by varying the core size in a drawn fiber, varying the index of refraction, and inducing stresses in the fiber.
There are known solutions that specify that the cladding-to-buffer boundary is a significant one and is included in the acoustic mode simulations, adding a set of boundary conditions to the set defined by the above boundary equations. This results in a waveguide consisting of hundreds of modal solutions greatly encumbering simulations of a fiber under design. More importantly, however, is that these solutions lead to a propagation constant β that is real-valued, thereby precluding the design of acoustically anti-guiding optical fibers, or fibers with large acoustic attenuation coefficients.
Thus, since acoustic anti-guidance does not exist according to known theory, this leads to designs that explicitly require the tailoring of the properties of acoustically waveguiding layers in order to suppress SBS. In the case of fibers where the core acoustic velocity is greater than that of the cladding, known teachings state that the core is dominated by an interaction with cladding modes that cannot be overcome. Therefore, known theories teach that SBS suppression via acoustically anti-guiding fibers with large acoustic waveguide attenuation coefficients is not possible.